sqrt{(x-2)^2+(y-2)^2}=sqrt{(x-(-2))^2+(y-4)^2} eching

$$\sqrt { ( x - 2 ) ^ { 2 } + ( y - 2 ) ^ { 2 } } = \sqrt { ( x - ( - 2 ) ) ^ { 2 } + ( y - 4 ) ^ { 2 } }$$

x uchun yechish

$$x=\frac{y-3}{2}$$

Yechim qadamlari

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(\sqrt{\left(x-2\right)^{2}+\left(y-2\right)^{2}}\right)^{2}=\left(\sqrt{\left(x-\left(-2\right)\right)^{2}+\left(y-4\right)^{2}}\right)^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(x-2\right)^{2} kengaytirilishi uchun ishlating.

$$\left(\sqrt{x^{2}-4x+4+\left(y-2\right)^{2}}\right)^{2}=\left(\sqrt{\left(x-\left(-2\right)\right)^{2}+\left(y-4\right)^{2}}\right)^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(y-2\right)^{2} kengaytirilishi uchun ishlating.

$$\left(\sqrt{x^{2}-4x+4+y^{2}-4y+4}\right)^{2}=\left(\sqrt{\left(x-\left(-2\right)\right)^{2}+\left(y-4\right)^{2}}\right)^{2}$$

8 olish uchun 4 va 4'ni qo'shing.

$$\left(\sqrt{x^{2}-4x+8+y^{2}-4y}\right)^{2}=\left(\sqrt{\left(x-\left(-2\right)\right)^{2}+\left(y-4\right)^{2}}\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{x^{2}-4x+8+y^{2}-4y} ga hisoblang va x^{2}-4x+8+y^{2}-4y ni qiymatni oling.

$$x^{2}-4x+8+y^{2}-4y=\left(\sqrt{\left(x-\left(-2\right)\right)^{2}+\left(y-4\right)^{2}}\right)^{2}$$

-2 ning teskarisi 2 ga teng.

$$x^{2}-4x+8+y^{2}-4y=\left(\sqrt{\left(x+2\right)^{2}+\left(y-4\right)^{2}}\right)^{2}$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(x+2\right)^{2} kengaytirilishi uchun ishlating.

$$x^{2}-4x+8+y^{2}-4y=\left(\sqrt{x^{2}+4x+4+\left(y-4\right)^{2}}\right)^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(y-4\right)^{2} kengaytirilishi uchun ishlating.

$$x^{2}-4x+8+y^{2}-4y=\left(\sqrt{x^{2}+4x+4+y^{2}-8y+16}\right)^{2}$$

20 olish uchun 4 va 16'ni qo'shing.

$$x^{2}-4x+8+y^{2}-4y=\left(\sqrt{x^{2}+4x+20+y^{2}-8y}\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{x^{2}+4x+20+y^{2}-8y} ga hisoblang va x^{2}+4x+20+y^{2}-8y ni qiymatni oling.

$$x^{2}-4x+8+y^{2}-4y=x^{2}+4x+20+y^{2}-8y$$

Ikkala tarafdan x^{2} ni ayirish.

$$x^{2}-4x+8+y^{2}-4y-x^{2}=4x+20+y^{2}-8y$$

0 ni olish uchun x^{2} va -x^{2} ni birlashtirish.

$$-4x+8+y^{2}-4y=4x+20+y^{2}-8y$$

Ikkala tarafdan 4x ni ayirish.

$$-4x+8+y^{2}-4y-4x=20+y^{2}-8y$$

-8x ni olish uchun -4x va -4x ni birlashtirish.

$$-8x+8+y^{2}-4y=20+y^{2}-8y$$

Ikkala tarafdan 8 ni ayirish.

$$-8x+y^{2}-4y=20+y^{2}-8y-8$$

12 olish uchun 20 dan 8 ni ayirish.

$$-8x+y^{2}-4y=12+y^{2}-8y$$

Ikkala tarafdan y^{2} ni ayirish.

$$-8x-4y=12+y^{2}-8y-y^{2}$$

0 ni olish uchun y^{2} va -y^{2} ni birlashtirish.

$$-8x-4y=12-8y$$

4y ni ikki tarafga qo’shing.

$$-8x=12-8y+4y$$

-4y ni olish uchun -8y va 4y ni birlashtirish.

$$-8x=12-4y$$

Ikki tarafini -8 ga bo‘ling.

$$\frac{-8x}{-8}=\frac{12-4y}{-8}$$

-8 ga bo'lish -8 ga ko'paytirishni bekor qiladi.

$$x=\frac{12-4y}{-8}$$

12-4y ni -8 ga bo'lish.

$$x=\frac{y-3}{2}$$

\sqrt{\left(x-2\right)^{2}+\left(y-2\right)^{2}}=\sqrt{\left(x-\left(-2\right)\right)^{2}+\left(y-4\right)^{2}} tenglamasida x uchun \frac{y-3}{2} ni almashtiring.

$$\sqrt{\left(\frac{y-3}{2}-2\right)^{2}+\left(y-2\right)^{2}}=\sqrt{\left(\frac{y-3}{2}-\left(-2\right)\right)^{2}+\left(y-4\right)^{2}}$$

Qisqartirish. x=\frac{y-3}{2} tenglamani qoniqtiradi.

$$\frac{1}{2}\left(65-30y+5y^{2}\right)^{\frac{1}{2}}=\frac{1}{2}\left(65-30y+5y^{2}\right)^{\frac{1}{2}}$$

\sqrt{\left(x-2\right)^{2}+\left(y-2\right)^{2}}=\sqrt{\left(y-4\right)^{2}+\left(x-\left(-2\right)\right)^{2}} tenglamasi noyob yechimga ega.

$$x=\frac{y-3}{2}$$

y uchun yechish

$$y=2x+3$$

Yechim qadamlari

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(\sqrt{\left(x-2\right)^{2}+\left(y-2\right)^{2}}\right)^{2}=\left(\sqrt{\left(x-\left(-2\right)\right)^{2}+\left(y-4\right)^{2}}\right)^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(x-2\right)^{2} kengaytirilishi uchun ishlating.

$$\left(\sqrt{x^{2}-4x+4+\left(y-2\right)^{2}}\right)^{2}=\left(\sqrt{\left(x-\left(-2\right)\right)^{2}+\left(y-4\right)^{2}}\right)^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(y-2\right)^{2} kengaytirilishi uchun ishlating.

$$\left(\sqrt{x^{2}-4x+4+y^{2}-4y+4}\right)^{2}=\left(\sqrt{\left(x-\left(-2\right)\right)^{2}+\left(y-4\right)^{2}}\right)^{2}$$

8 olish uchun 4 va 4'ni qo'shing.

$$\left(\sqrt{x^{2}-4x+8+y^{2}-4y}\right)^{2}=\left(\sqrt{\left(x-\left(-2\right)\right)^{2}+\left(y-4\right)^{2}}\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{x^{2}-4x+8+y^{2}-4y} ga hisoblang va x^{2}-4x+8+y^{2}-4y ni qiymatni oling.

$$x^{2}-4x+8+y^{2}-4y=\left(\sqrt{\left(x-\left(-2\right)\right)^{2}+\left(y-4\right)^{2}}\right)^{2}$$

-2 ning teskarisi 2 ga teng.

$$x^{2}-4x+8+y^{2}-4y=\left(\sqrt{\left(x+2\right)^{2}+\left(y-4\right)^{2}}\right)^{2}$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(x+2\right)^{2} kengaytirilishi uchun ishlating.

$$x^{2}-4x+8+y^{2}-4y=\left(\sqrt{x^{2}+4x+4+\left(y-4\right)^{2}}\right)^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(y-4\right)^{2} kengaytirilishi uchun ishlating.

$$x^{2}-4x+8+y^{2}-4y=\left(\sqrt{x^{2}+4x+4+y^{2}-8y+16}\right)^{2}$$

20 olish uchun 4 va 16'ni qo'shing.

$$x^{2}-4x+8+y^{2}-4y=\left(\sqrt{x^{2}+4x+20+y^{2}-8y}\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{x^{2}+4x+20+y^{2}-8y} ga hisoblang va x^{2}+4x+20+y^{2}-8y ni qiymatni oling.

$$x^{2}-4x+8+y^{2}-4y=x^{2}+4x+20+y^{2}-8y$$

Ikkala tarafdan y^{2} ni ayirish.

$$x^{2}-4x+8+y^{2}-4y-y^{2}=x^{2}+4x+20-8y$$

0 ni olish uchun y^{2} va -y^{2} ni birlashtirish.

$$x^{2}-4x+8-4y=x^{2}+4x+20-8y$$

8y ni ikki tarafga qo’shing.

$$x^{2}-4x+8-4y+8y=x^{2}+4x+20$$

4y ni olish uchun -4y va 8y ni birlashtirish.

$$x^{2}-4x+8+4y=x^{2}+4x+20$$

Ikkala tarafdan x^{2} ni ayirish.

$$-4x+8+4y=x^{2}+4x+20-x^{2}$$

0 ni olish uchun x^{2} va -x^{2} ni birlashtirish.

$$-4x+8+4y=4x+20$$

4x ni ikki tarafga qo’shing.

$$8+4y=4x+20+4x$$

8x ni olish uchun 4x va 4x ni birlashtirish.

$$8+4y=8x+20$$

Ikkala tarafdan 8 ni ayirish.

$$4y=8x+20-8$$

12 olish uchun 20 dan 8 ni ayirish.

$$4y=8x+12$$

Ikki tarafini 4 ga bo‘ling.

$$\frac{4y}{4}=\frac{8x+12}{4}$$

4 ga bo'lish 4 ga ko'paytirishni bekor qiladi.

$$y=\frac{8x+12}{4}$$

8x+12 ni 4 ga bo'lish.

$$y=2x+3$$

\sqrt{\left(x-2\right)^{2}+\left(y-2\right)^{2}}=\sqrt{\left(x-\left(-2\right)\right)^{2}+\left(y-4\right)^{2}} tenglamasida y uchun 2x+3 ni almashtiring.

$$\sqrt{\left(x-2\right)^{2}+\left(2x+3-2\right)^{2}}=\sqrt{\left(x-\left(-2\right)\right)^{2}+\left(2x+3-4\right)^{2}}$$

Qisqartirish. y=2x+3 tenglamani qoniqtiradi.

$$\left(5+5x^{2}\right)^{\frac{1}{2}}=\left(5+5x^{2}\right)^{\frac{1}{2}}$$

\sqrt{\left(x-2\right)^{2}+\left(y-2\right)^{2}}=\sqrt{\left(y-4\right)^{2}+\left(x-\left(-2\right)\right)^{2}} tenglamasi noyob yechimga ega.

$$y=2x+3$$